Finite Group Databasebeta

← #30670850

Subgroups — #30670850

Order 16768.

800 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 161 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862214C211yesmaximal, characteristic
2C2×D4×Dic131C211yesmaximal, characteristic, complemented
3C22×(C2×D131)⋊C4C211yesmaximal, characteristic, complemented
4#22553234C212yesmaximal, complemented
5#22553227C212yesmaximal, complemented
6#22553202C212yesmaximal, complemented
7#22553189C212yesmaximal, complemented
8#22553175C211yesmaximal, characteristic, complemented
9#22553170C211yesmaximal, characteristic, complemented
10#22553167C211yesmaximal, characteristic, complemented
11C2×#123×C131C211yesmaximal, characteristic, complemented
12C24×D131C2211yescharacteristic
13C22×C22⋊D131C2212yes—
14C23×Dic131C2211yescharacteristic
15C23×Dic131C2211yescharacteristic, complemented
16#16862214—24no—
17D4×Dic131—24no—
18C2×C4⋊Dic131C2211yescharacteristic, complemented
19C2×Dic131⋊C4C2212yescomplemented
20C2×C22⋊Dic131C2211yescharacteristic
21C2×C22⋊Dic131C2211yescharacteristic
22C2×C22⋊Dic131C2211yescharacteristic, complemented
23C2×(C2×D131)⋊C4—24no—
24C2×(C2×D131)⋊C4C2212yes—
25C2×(C2×D131)⋊C4C2212yes—
26C2×(C2×D131)⋊C4C2212yes—
27C2×(C2×D131)⋊C4C2212yes—
28C2×(C2×D131)⋊C4C2212yes—
29C2×C4×Dic131C2211yescharacteristic, complemented
30C131⋊#95C2212yescomplemented
31C131⋊#95C2212yescomplemented
32C131⋊#95C2212yescomplemented
33C22×D4×C131C2211yescharacteristic
34C23×C4×C131C2211yescharacteristic
35#123×C131—24no—
36C2×C4⋊C4×C131C2212yescomplemented
37C2×C22⋊C4×C131C2211yescharacteristic
38C2×C22⋊C4×C131C2212yescomplemented
39C23×D131—24no—
40C23×D131—22no—
41C23×D131—22no—
42C23×D131—22no—
43C23×D131—42no—
44C23×D131C2312yes—
45C2×C22⋊D131—24no—
46C2×C22⋊D131—24no—
47C2×C22⋊D131—42no—
48C22×Dic131—22no—
49C22×Dic131—42no—
50C22×Dic131—42no—
51C22×Dic131D414yescomplemented
52C22×Dic131C2311yescharacteristic
53C22×Dic131C2312yes—
54C22×Dic131C2311yescharacteristic
55C22×Dic131C2311yescharacteristic
56C22×Dic131C2311yescharacteristic
57C4⋊Dic131—22no—
58Dic131⋊C4—42no—
59C22⋊Dic131—22no—
60C22⋊Dic131—22no—
61C22⋊Dic131—22no—
62(C2×D131)⋊C4—24no—
63(C2×D131)⋊C4—24no—
64(C2×D131)⋊C4—24no—
65(C2×D131)⋊C4—24no—
66(C2×D131)⋊C4—42no—
67C4×Dic131—22no—
68C24×C131C2311yescharacteristic
69C24×C131C2311yescharacteristic
70C2×D4×C131—22no—
71C2×D4×C131—22no—
72C2×D4×C131D414yes—
73C22×C4×C131—22no—
74C22×C4×C131—22no—
75C22×C4×C131—22no—
76C22×C4×C131C2312yes—
77C22×C4×C131C2312yes—
78C22×C4×C131C2311yescharacteristic
79C4⋊C4×C131—24no—
80C22⋊C4×C131—22no—
81C22⋊C4×C131—24no—
82C22×D131—28no—
83C22×D131—22no—
84C22×D131—22no—
85C22×D131—22no—
86C22×D131—22no—
87C22×D131—22no—
88C22×D131—22no—
89C22×D131—48no—
90C22×D131—48no—
91C22×D131—44no—
92C22×D131—44no—
93C22×D131#4114yes—
94C22⋊D131—48no—
95C2×Dic131—22no—
96C2×Dic131—22no—
97C2×Dic131—21no—
98C2×Dic131—22no—
99C2×Dic131—21no—
100C2×Dic131—22no—
101C2×Dic131—22no—
102C2×Dic131—22no—
103C2×Dic131—22no—
104C2×Dic131—44no—
105C2×Dic131—44no—
106C2×Dic131C2×D412yes—
107C2×Dic131C2×D412yes—
108C23×C131—22no—
109C23×C131—22no—
110C23×C131—21no—
111C23×C131—21no—
112C23×C131—22no—
113C23×C131—22no—
114C23×C131C2×D412yes—
115C23×C131C2×D412yes—
116C23×C131C2×D412yes—
117C23×C131#4112yes—
118C23×C131C2411yescharacteristic
119D4×C131—28no—
120C2×C4×C131—21no—
121C2×C4×C131—21no—
122C2×C4×C131—22no—
123C2×C4×C131—22no—
124C2×C4×C131—22no—
125C2×C4×C131—22no—
126C2×C4×C131—22no—
127C2×C4×C131—22no—
128C2×C4×C131—44no—
129C2×C4×C131C2×D412yes—
130C22×C131—22no—
131C22×C131—22no—
132C22×C131—24no—
133C22×C131—24no—
134C22×C131—22no—
135C22×C131—24no—
136C22×C131—41no—
137C22×C131—41no—
138C22×C131—42no—
139C22×C131#12114yes—
140C22×C131C22×D411yescharacteristic
141C22×C131C22×D411yescharacteristic
142C22×C131C2×#4112yes—
143C22×C131C2×#4111yescharacteristic
144C22×C131H(5,𝔽2)12yes—
145C2×D131—24no—
146C2×D131—24no—
147C2×D131—24no—
148C2×D131—416no—
149C2×D131—42no—
150C2×D131—42no—
151C2×D131—42no—
152C2×D131—42no—
153C4×C131—22no—
154C4×C131—42no—
155C4×C131—42no—
156Dic131—22no—
157Dic131—22no—
158Dic131—41no—
159Dic131—41no—
160Dic131—42no—
161C2×C131—24no—
162C2×C131—42no—
163C2×C131C2×#12111yescharacteristic
164C2×C131#52511yescharacteristic
165C2×C131#53411yescharacteristic
166C2×C131#54612yes—
167C2×C131#55112yes—
168D131—44no—
169C131#253411yescharacteristic, complemented, Sylow
170#2534—1311nomaximal, Sylow
171C2×#123D13111yescharacteristic, complemented
172C2×#120—1311no—
173C2×C4×D4—1311no—
174C22×C22⋊C4—1311no—
175#396—1312no—
176#394—1312no—
177#388—1312no—
178#386—1312no—
179#381—1311no—
180#380—1311no—
181#379—1311no—
182C25—1311no—
183C22×D4—1312no—
184C22×D4C2×D13111yescharacteristic
185C23×C4—1311no—
186C23×C4—1311no—
187C23×C4C2×D13111yescharacteristic
188#123—24no—
189#120—2624no—
190C4×D4—2624no—
191C2×C4⋊C4—1311no—
192C2×C4⋊C4—1312no—
193C2×C4⋊C4C2×D13112yescomplemented
194C2×C22⋊C4—2624no—
195C2×C22⋊C4—1311no—
196C2×C22⋊C4—1311no—
197C2×C22⋊C4—1311no—
198C2×C22⋊C4—1312no—
199C2×C22⋊C4—1312no—
200C2×C22⋊C4—1312no—
201C2×C22⋊C4—1312no—
202C2×C22⋊C4—1312no—
203C2×C22⋊C4C2×D13111yescharacteristic
204C2×C22⋊C4C2×D13112yescomplemented
205C2×C42—1311no—
206#95—1312no—
207#95—1312no—
208#95—1312no—
209C24—2624no—
210C24—2622no—
211C24—2622no—
212C24—2622no—
213C24—5242no—
214C24—1312no—
215C24C22×D13111yescharacteristic
216C24C22×D13111yescharacteristic
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—2624no—
220C2×D4—2624no—
221C2×D4—5242no—
222C2×D4C22⋊D13114yes—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—2622no—
227C22×C4—5242no—
228C22×C4—5242no—
229C22×C4—1312no—
230C22×C4—1311no—
231C22×C4—1311no—
232C22×C4—1311no—
233C22×C4—1311no—
234C22×C4—1314no—
235C22×C4C22×D13112yes—
236C22×C4C22×D13112yes—
237C22×C4C22×D13111yescharacteristic
238C4⋊C4—24no—
239C4⋊C4—2622no—
240C4⋊C4—5242no—
241C22⋊C4—22no—
242C22⋊C4—24no—
243C22⋊C4—2624no—
244C22⋊C4—2624no—
245C22⋊C4—2624no—
246C22⋊C4—2622no—
247C22⋊C4—2622no—
248C22⋊C4—2622no—
249C22⋊C4—2624no—
250C22⋊C4—5242no—
251C42—2622no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—22no—
256C23—21no—
257C23—22no—
258C23—2628no—
259C23—2622no—
260C23—2622no—
261C23—2622no—
262C23—2622no—
263C23—2622no—
264C23—2622no—
265C23—5248no—
266C23—5248no—
267C23—5244no—
268C23—5244no—
269C23—1314no—
270C23D4×D13112yes—
271C23C2×C22⋊D13112yes—
272C23C2×C22⋊D13112yes—
273C23#1452351212yes—
274C23C23×D13111yescharacteristic
275D4—28no—
276D4—5248no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—44no—
286C2×C4—2622no—
287C2×C4—2622no—
288C2×C4—2621no—
289C2×C4—2621no—
290C2×C4—2622no—
291C2×C4—2622no—
292C2×C4—2622no—
293C2×C4—2622no—
294C2×C4—2622no—
295C2×C4—5244no—
296C2×C4—5244no—
297C2×C4—1312no—
298C2×C4—1312no—
299C2×C4C2×C22⋊D13112yes—
300C22—22no—
301C22—24no—
302C22—24no—
303C22—22no—
304C22—24no—
305C22—22no—
306C22—41no—
307C22—42no—
308C22—41no—
309C22—2624no—
310C22—2624no—
311C22—2624no—
312C22—52416no—
313C22—5242no—
314C22—5242no—
315C22—5242no—
316C22—5242no—
317C22#1686221814yes—
318C22C2×D4×D13111yescharacteristic
319C22C22×C22⋊D13111yescharacteristic
320C22#41×D13112yes—
321C22C2×#1452351211yescharacteristic
322C22#1686229212yes—
323C4—22no—
324C4—42no—
325C4—42no—
326C4—2622no—
327C4—2622no—
328C4—5241no—
329C4—5242no—
330C4—5241no—
331C2—24no—
332C2—42no—
333C2—5244no—
334C2C2×#1686221811yescharacteristic
335C2#2255382011yescharacteristic
336C2#2255387311yescharacteristic
337C2#2255395012yes—
338C2#2255398512yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC2×#123×C131C2
prime radicalC2×#16862214C2
hypercenterC2×#123D131
hypocenterC131#2534
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×#123

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×D4×Dic131C2semidirect
3. C22×(C2×D131)⋊C4C2semidirect
4. #22553234C2semidirect
5. #22553227C2semidirect
6. #22553202C2semidirect
7. #22553189C2semidirect
8. #22553175C2semidirect
9. #22553170C2semidirect
10. #22553167C2semidirect
11. C2×#123×C131C2semidirect
15. C23×Dic131C22semidirect
18. C2×C4⋊Dic131C22semidirect
19. C2×Dic131⋊C4C22semidirect
22. C2×C22⋊Dic131C22semidirect
29. C2×C4×Dic131C22semidirect
30. C131⋊#95C22semidirect
31. C131⋊#95C22semidirect
32. C131⋊#95C22semidirect
36. C2×C4⋊C4×C131C22semidirect
38. C2×C22⋊C4×C131C22semidirect
51. C22×Dic131D4semidirect
169. C131#2534semidirect
171. C2×#123D131semidirect
193. C2×C4⋊C4C2×D131semidirect
204. C2×C22⋊C4C2×D131semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168622142
C2×D4×Dic1312
C22×(C2×D131)⋊C42
#225532342
#225532272
#225532022
#225531892
#225531752
#225531702
#225531672
C2×#123×C1312
#2534131